Partial regularity under anisotropic \((p,q)\) growth conditions. (English) Zbl 0807.49010

Let \(\Omega\) be an open subset of \(\mathbb{R}^ n\), \(f: \mathbb{R}^{n\cdot N}\to \mathbb{R}\) be a function of class \(C^ 2\) satisfying the inequality \(|\xi|^ p\leq f(\xi)\leq C(1+ |\xi|^ q)\) for any \(\xi\in \mathbb{R}^{n\cdot N}\), and set \[ F(u)= \int_ \Omega f(Du)dx \] for any \(u\in W^{1,p}(\Omega,\mathbb{R}^ N)\).
The regularity of minimizers of integral functionals of the kind (1) under anisotropic \((p,q)\) growth conditions of \(f\) is studied. In the particular case of \[ F(u)= \int_ \Omega\left[| Du|^ p+ \sum^ k_{\alpha=1} | D_ \alpha u|^{p_ \alpha}\right] dx,\tag{2} \] where \(1\leq k\leq n\) and \(2\leq p< p_ \alpha\) for \(\alpha= 1,\dots,k\), the following result is proved.
Theorem. If \(u\in W^{1,p}(\Omega,\mathbb{R}^ N)\) is minimizer of (2), with \(D_ \alpha u\in L^{p_ \alpha}(\Omega)\) for \(\alpha= 1,\dots,k\), and if \(\max\{p_ \alpha: \alpha= 1,\dots,k\}<\bar p^*\), where \(\bar p^*= n\bar p/(n- \bar p)\), \(\bar p^{-1}= n^{-1}\left[(n- k) p^{- 1}+ \sum^ k_{\alpha=1} p^{-1}_ \alpha\right]\), then \(Du\) is Hölder continuous in an open set \(\Omega_ 0\) such that \(\text{meas}(\Omega\backslash\Omega_ 0)= 0\).


49J40 Variational inequalities
35J50 Variational methods for elliptic systems
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